Electrokinetics - COnnecting REpositories · s 14.GS: CIR263 c.1 STATEOFILLINOIS...

s

14.GS:

CIR263c. 1

STATE OF ILLINOIS

WILLIAM G. STRATTON, Governor

DEPARTMENT OF REGISTRATION AND EDUCATIONVERA M.BINKS, Director

ELECTROKINETICSI. — Electroviscosity and the

Flow of Reservoir Fluids

Norman Street

DIVISION OF THE

ILLINOIS STATE GEOLOGICAL SURVEYJOHN C. FRYE, Chief URBANA

CIRCULAR 263 1959

Digitized by the Internet Archive

in 2012 with funding from

University of Illinois Urbana-Champaign

http://archive.org/details/electrokinetics263stre

ELECTROKINETICS

I. — Electroviscosity and the Flow of Reservoir Fluids

Norman Street

ABSTRACT

Liquids flowing through narrow capillaries frequently exhibit a

viscosity that is higher than normal; usually it is high for low-conduct-

ivity liquids but is low for very conductive liquids or concentrated so-

lutions. This effect may have some influence on the flow of oil andwater through petroleum reservoirs, although little note has yet beentaken of the possibility.

The theory underlying this electroviscous effect is simply devel-

oped and some calculations as to its order of magnitude are presented.

An equation is given for the electroviscous effect in two-phase flow

through a "Yuster" model.

INTRODUCTION

Experimental observations of an increase in the viscosity of a liquid flowing

through a narrow capillary have been reported by a number of workers (Reekie andAird, 1945; Terzaghi, 1931; Macauly, 1936; Henniker, 1952), and a recent workerhas published a theoretical analysis of the origin of such increase in viscosity

(Elton, 1948).

The electrical origin of the effect was early recognized. In fact, Smoluchow-ski (1916) had given an equation for the "electroviscous" effect in suspensionflow; Bull (1932) gave a simple derivation of the effect for flow through capillaries;

and White, Monaghan, and Urban (1935) discussed the effect of electrical chargeson flow through cellophane membranes. More recently, Lorenz (1952), in a sophis-

ticated approach to electrokinetic phenomena, dealt with electroviscosity of liquid

flow through both capillaries and porous plugs.

The effect is apparent only for flow through very small capillaries when it is

controlled by the charge at the solid-liquid interface (perhaps more correctly the

zeta potential), the conductivity of the flowing liquid, and its dielectric constant.

We shall see that high interfacial potential and low liquid conductivity are the

factors that contribute most to high apparent viscosity.

If an aqueous solution is relatively concentrated its conductivity is high,

and even though the solid surface has a high charge, the interfacial or zeta poten-tial nevertheless will be low because the double layer (vide infra) is compressedand there will be no discernable electroviscous effect. If the aqueous solution is

dilute, however, conductivity is much lower, zeta potentials generally are high,

and a real electroviscous effect may exist. For example, Reekie and Aird (1945)

report five to ten times the normal bulk viscosity for water flowing through dia-

phragms of rouge, French chalk, and carborundum.In non-aqueous systems we can expect that the conductivity normally will

be very low, but, on the other hand, the zeta potentials may also be low although

[1]

2 ILLINOIS STATE GEOLOGICAL SURVEY

there is some evidence that this is not always true (van der Minne and Hermanie,1953). If the zeta potentials have some reasonable value coupled with very lowconductivity, the electroviscous effect may be large.

Of course the flow of reservoir fluids is not generally a matter of simplesingle-phase flow. But even if oil alone were flowing the effect would be compli-cated because the mineral surfaces are usually water wet and therefore, the oil

would not be in direct contact with the solid surface.

An immobile water film may exist, thick enough and conductive enough to

provide a path for the back flow of any current generated, without such current

flow affecting the fluid flow of the hydrocarbon. Or it is possible for a water film

to be much thicker and to be flowing along with the oil. In either case the problemis further complicated by the fact that there will be set up not only a solid-water

potential but also an oil-water potential which must be taken into account in con-sidering the final effect.

In the following discussion some attempt is made to investigate these pos-sibilities and to indicate the order of magnitude of the effects themselves. As the

concepts of zeta potential, streaming potential, and electro-osmosis may be some-what unfamiliar, a small space is first devoted to a discussion of the origin of

surface charge and potential and to simple derivations of the equations for stream-ing potential and electro-osmosis.

BASIC PRINCIPLES

Surface Charge

Mineral surfaces in contact with aqueous solutions are almost invariably

charged. The most important charging mechanisms are dissociation of ionogenicgroups and the preferential adsorption of one ion from the solution (Mukherjee,

1920). The electrical potential resulting from these charges is called the zeta

potential

.

Inasmuch as Coulomb attraction exists between the charged surface and anyoppositely charged ions in the solution, it may seem surprising that the surface

remains charged rather than being immediately neutralized by combination with op-positely charged ions from the solution. In order to understand this, it is neces-sary to consider the forces existing between simple dissolved ions.

Consider the case of two oppositely charged univalent ions, for example,

Na+ and Cl~ in aqueous solution. The attractive force between them is

(1)

Dx 2

where

e = electronic charge (4.8 x 10" 10 e.s.u.)

D = dielectric constant (80 for water)

x = distance separating the centers of the ions

However, as is well known, such ions do not coalesce by the operation of

such a force because although the Coulomb attraction between unlike ions tends to

draw them together, thermal (Brownian) motions tend to distribute them throughout

the solution.

Let us compare the thermal energy of simple ions with the energy necessaryto separate them. The radii of hydrated Na+ and CI" ions are 2.5 and 2.0 A,

FLOW OF RESERVOIR FLUIDS

respectively, so that at their closest distance of approach the charges are sepa-

rated by 4.5 A. So by equation (1)

(4.8 x lCT 10 )2

, . ...fi

.

= -* r— = 1.4x10 D dyne80(4.5 x 10" 8

)

2

To obtain a value for the work necessary to separate these ions we must in-

tegrate force times distance from x = r to x = oo, that is

f e+e" e+e~ (4.8 x 10" 10)2 _ . lf1-i4=

J 7 dx =~n7~

=ft "

x { rDx z ur 80(4.5 x 10"°)

The thermal energy of a molecule or ion is given by kinetic theory and is

Kinetic energy = 3/2 kT

wherek = Boltzmann's constant (1.37 x 10

-1 erg/molec./deg.)

T = absolute temperature

Thus at 20°C

Kinetic energy = 6.10 x 10 erg

So we see that in water, the thermal energy of the separated ions and the

energy necessary to separate them are very nearly the same. In solvents with

dielectric constants smaller than it is in water, the force attracting the ions andthe energy necessary to separate them will be much greater, and such ions are

not dissociated in solution.

Similar considerations apply to the ions adsorbed on the mineral surfaces

and the oppositely charged ions in the solution surrounding the surface.

Electric Potential

The work necessary to bring together from infinity two ions of opposite sign

has the same magnitude as that necessary to separate them to infinity from their

distance of closest approach. It is convenient to define the electric potential as

the work required to bring unit charge from infinity to a charged point of like sign,

or alternatively, as the work released when a unit charge of unlike sign is brought

to this point from infinity.

The potential function is a property of the space surrounding electric charges,

every point in space has a potential due to the presence of the ion, and if there

are other ions in the space, the total potential at any point is given by the alge-

braic sum of the individual potentials at that point due to each ion. The work ne-cessary to bring unit charge from infinity to a distance r from the center of an ion

is equal to e/Dr, and this is the potential at a distance r.

Zeta Potential and Double Layer Thickness

The charged particle surface attracts water dipoles and is covered by a layer

of strongly bound water molecules that become part of the kinetic unit. Trappedamong the water molecules are commonly some positive charges that also becomepart of the kinetic unit and by their presence reduce the net charge on the particle

(fig. 1).


When there is relative

movement between the particle

and the liquid, the plane of shear

is at the outermost edge of the

solvated layer, and so it is the

net charge that is important in

electrokinetic phenomena . Thezeta potential then is determinedby the work necessary to bring

unit charge from infinity to the

surface of shear.

Surrounding the particle,

but relatively distant from it, is

an atmosphere of ions in constant

thermal movement. The numberof positive ions (assuming the

particle surface to be negative)

in this atmosphere is greater

than the number of negative ions,

and there are enough positive

ions, on a time average, to bal-

ance out the net negative chargeon the particle. The ions of the

ionic atmosphere form the "dif-

fuse double layer. " They are not immobilized by the Coulomb attraction of the

particle but constantly move in and out between the double layer and the main bodyof the liquid. It is convenient to consider that the excess positive charges are

on a concentric shell at a fixed distance from the particle, the shell is the electri-

cal "center of gravity" of the ion cloud, and the distance from the surface of shear

to the shell is the thickness of the double layer.

If a suspended charged particle is subjected to an electric field it moves to

one or the other of the electrodes, at the same time the oppositely charged ionic

atmosphere tends to move in the opposite direction and consequently to retard the

motion of the particle. The distance from the surface of shear to the hypothetical

concentric shell of oppositely charged ions is chosen so that if the ions were ac-tually on this shell they would have the same retarding effect as the ion atmosphere.

Thus, although we assume that the opposite charges are present only on the sur-

face of the shell, nevertheless we can feel confident that their effect is the sameas when they are scattered through the atmosphere.

With this model, a large, non-conducting particle, together with its doublelayer, constitutes a parallel plate condenser with its plates separated by a dis-

tance X, the "thickness" of the double layer. In the next two sections we shall

examine (a) the effect on zeta of varying the distance of separation of two suchplates (at a fixed surface charge density), and (b) the effect of concentration andtype of electrolyte in solution, on the double layer thickness. The two taken to-

gether show the effect of concentration on zeta at constant surface charge density.

Fig. 1. - A charged spherical particle and its

bound water molecules.

Effect of Double Layer Thickness on the Zeta Potential

Consider a particle of radius r and charge Q surrounded by a concentric

shell of radius (r + X ) and charge -Q (fig. 2).


The potential at the surface of the

sphere is Q/Dr, this being the work ne-

cessary to bring unit charge of like sign

from infinity to a distance r from the

center of the sphere. The resultant po-

tential of a condenser consisting of twosuch concentric spheres is the algebraic

sum of the potentials due to the inner

sphere at its surface and the outer

sphere at the surface of the inner sphere.

The potential on the surface of the

inner sphere in the absence of the outer

sphere would be Q/Dr; the potential dueto the outer sphere at any point inside

it is -Q/D(r + X ); therefore,

C- Dr Dr

X

r + XD(r + X)

and since X<<r

£ = QX/Dr2

If the surface charge density is a, then Q = 4TTr2crand

£ = 4ttctX/D

Fig. 2. - A charged spherical particle andits associated double layer.

(2)

Although the latter equation has been obtained by considering a spherical

particle, it is generally applicable and the potential difference between two paral-

lel plates of surface charge density & when separated by a distance X in a mediumof dielectric constant D is also given by equation (2).

Effect of Concentration on Double Layer Thickness

As we have seen, the double layer is actually diffuse, and the ions forming

it are not fixed at any one distance away from the particle surface, but form anatmosphere around it. There is competition between the thermal Brownian move-ments tending to distribute the ions evenly throughout the solution and the attrac-

tive forces of the charged surface drawing unlike ions to it. Like ions tend to bedriven away from the surface by repulsive forces, and a large negatively chargedparticle will have more positive than negative ions near its surface; nevertheless,

in an element of volume remote from the surface, the number of positive and nega-tive ions will be equal.

If the resultant potential due to the charged surface and the ionic atmosphereis ty , energy equal to e V is released when a positive ion is brought up from the

main body of the solution to a point of potential ^ , and an amount of energy equalto -e'/'is required to bring a negative ion to the same point.

When equilibrium is established, by Boltzmann's principle the number of

negative ions (n_) per unit volume is

,-e^/kTn_ = ne

where n is the total number of ions per cc. in the bulk solution.

number of positive ions {n+) is .

n+ = ne+eVykT

Similarly the

ILLINOIS STATE GEOLOGICAL SURVEY

The charge density (p) in

unit volume of potential

<//, is

p = (n_ - n+)e =

(e-e</>/kT_ e + e^/kT) ne =

-2ne sinh e^/kT

and if the potential is

small so that e<///kT<< 1,

sinh e r/kT can be replaced

by e^/kl, and so

700

600

500

=> 400E

p = -2ne 2 <///kT (3)

Pois son's equation

states that in every point

of the space charge

V 2^ = -4irp/D (4)

(V Laplace operator -

dx 2

a 2 a 2,+

Ody z dz 2

200

ISODIUM CHLORIDE

CALCIUM CHLORI

Normohty

Fig. 3. - Double-layer thickness plotted against con-

centration for calcium chloride and sodium chloride

solutions.

Combining equations (3) and (4) we get

V 2t = 8Trne 2/DkT • t (5)

DkTThe expression v

8-rvne 2 nas tne dimensions of a length and is set equal to

\/k and can be shown (see appendix) to be equal to X .

For the general case of multivalent ions

1A- 1000D RT8 e 2 N 2

u.

wheree = electronic charge

N = Avogadro's numberu. = ionic strength = 1/2^c^ 2

D = dielectric constant

c^ = cone, of ion i (mols/litre)

*i- valence of ion i

At 25 °C in water

x = 0.327 x lO 8^Figure 3 shows double-layer thickness plotted against normality for sodium

chloride and calcium chloride solutions.


STREAMING POTENTIAL

When liquid is forced through a capillary tube, a potential difference (the

streaming potential) may be produced between its ends.

Consider a tube of radius a and length 1 through which a liquid is caused to

flow by a pressure difference P. The force causing the liquid to flow is -n-a^P in

the direction of flow. Let the velocity of the liquid at a distance r from the axis

be v; the velocity gradient will be dv/dr so that the viscous force tending to retard

the flow i s 77 -dv/dr- 2iral

.

At equilibrium these two forces are equal so that

TTa2P =-77- dv/dr -2^1

Assume that the capillary wall is charged and that this surface charge hasassociated with it a double layer of thickness X , very small in comparison with

the radius of the capillary. The velocity of liquid within the double layer varies

linearly with the distance x from the wall and is zero at x = 0; dv/dx = -dv/dr canbe replaced by v/X , hence

TTa 2 P = +77 -v/X -2Tral or v = +X Pa/2177

Inasmuch as the surface charge density is cr/cm , the current of convection I is

IQ= 2ira-0--v = TTa

2 PXcr/77l

This movement of charge is opposed by the streaming potential difference (E) be-tween the ends of the capillary. The amount of charge carried back through the

area Tra2 by the liquid of conductivity K is

At equilibrium

Is

= Krra 2E/l

lo= I S

and therefore

E/P = XC/K77

Inasmuch as

C= D£/4rX

therefore

E = D£P/4ttKt7 (6)

In practical units, with water of viscosity 0.01 poise,

_ £d 13.6 x 981 x 80 x 300E = — — volts

K 300 x 9 x 10 11 x 4 x 3.14 x 0.01

ELECTRO-OSMOSIS

Capillary

We have seen that fluid forced through a capillary tube gives rise to a stream-

ing potential. Conversely, if a potential difference is applied across the tube it

will give rise to an electro-osmotic flow of the fluid. A simple derivation of the

velocity of electro-osmosis flow follows (Perrin, 1904; Butler, 1940).


Assume that the charge density of the double layer, which is free to move,is a" and that it is distant X from the surface. If the applied potential difference

has a strength E/unit length, an electrical force Eo'is imposed on the charge.The frictional force retarding flow isrjv/k and at equilibrium

Ecr =77 v/X

and so

v = E a X/77

and if

<f = d£/4ttX

therefore

v = ED£/4ttt7 (7)

If the counter pressure (P) developed by the electro-osmotic displacement is

measured, then this is determined for a single capillary of unit length by Poiseuille's

law

Vol = TrPr4/8'? = -rrr2 x ED£/4ttt7

or

P = 2DE£/rrr 2 (7a)

Equation (7) can be expressed differently by the following substitutions

E = ~—

,

V - -rrr2v

irr 2 K

Therefore

V = -irr2v = d£/4it77 x i/K (8)

Porous Plug

Smoluchowski (1903) showed that equation (8) is applicable to flow through

porous plugs. Then, if the measured cell constant of the plug is C with a solution

of conductivity K

i = KE/C

and so it follows that

V = d£/4ttt7 x E/C (9)

In order to express this in terms of an electro-osmotic pressure we mustequate it to Darcy's equation rather than to Poiseuille's. Thus

V = PkA/171 - D£/47rn x E/C

so that

P = D£E/47Tk x L/AC (9a)

where k = permeability and A = area of the plug.

FLOW OF RESERVOIR FLUIDS 9

ELECTROVISCOUS EFFECT

Capillary Flow

Consider, as before, a narrow tube with a relatively high surface charge

through which a liquid is flowing. A pressure drop, P, applied across the ends of

the tube causes the fluid flow. The flow gives rise to a streaming potential, (E),

as shown above, and the potential difference in turn gives rise to a back electro-

osmotic pressure, (P}).

Because of the back pressure, the pressure actually causing flow, (P2), is

less than the applied pressure, (P), by the back electro-endosmotic pressure Pi

(fig. 4), that is,

P2 = P - Pj

To determine viscosity by the Poiseuille equa-tion, the volume rate of flow through a tube is meas- p

\

* 1—

ured at some known pressure drop. For a narrow cap-illary this measurement gives a value for the viscosity

which is greater than the true value because the mag- Fig. 4. - Schematic diagram

nitude of the pressure drop used in the equation is of applied pressure, P, being

greater than that causing the flow. Thus reduced by opposing backpressure, P, .

7} a= PiTrV81v (10)

The true viscosity of the liquid is given by

17 - P2Tvr

2/81v (11)

where 7^ a = apparent viscosity and 17 = true viscosity, because the pressure P2

(less than P) is that which is actually instrumental in causing the fluid flow.

The streaming potential caused by the flowing liquid has a value of

E = P2d£/4it77K

so that

P2= 4it77KE/d£

This streaming potential causes an electro-osmotic back pressure of magni-tude P^ where

P, - 2D£EAa 2

Thus, because

From equations (10) and (11)

and so

P = P:+ P

2

P = 4ttt7KE/D£+ 2D£E/ira 2

T)a/7)

= P/P2

V a/V= 1 + D 2

£2/2Tr 2a 27?K (12)

Figure 5, which is a plot of £ in millivolts against {7ja -rj)K, shows how the

zeta potential, conductivity, and tube radius affect apparent viscosity in aqueoussolutions.


7

60

„ 50

2 40

Z 30

20

10

1.0 X I0"4

y

/0.2 X 104 /•0.I5X 10

4-5 ^^^

/

/ -b ^-~—

'

&z-.i i

OAXJO^

0.2 X IQ'j

1 x I0" 5

i i. i . i i l , : .

.02 .04 .06 .08 .10 .12

07 — 77 ) K X I06

.18 .20

Fig. 5. - (773 -77)K plotted against zeta potential

for various tube radii.

For example, with £ = 50 millivolts and a = 0.8 x 10~ 5, (Va -V)K- = 0.156 x

10-6 . Therefore, if K - 1 x 10~ 6, V a~V = 0.156 or7? a = f] + 0.156. If a = . 2 x

10" 4, (Va-V)K = 0.025 x 10- 6

, V a-V = 0.025 or?7 a = 7? + 0.025.

Flow Through Porous Plugs

We can apply similar reasoning to the flow of a liquid through a porous plug;

the streaming potential has the same value as before and so also therefore has P2

.

The electro-osmotic back pressure, (P-^), is now given by equation (9a) and so

P = P2 + Pi = 4Trr/KE/D£ + d£e/4tt)<: x L/AC

As before

and so

where

v a/v = P/P 2

V a/V = 1 + D 2C 2/l6u 2k^K x 1/F (13)

F = AC/L = Formation factor, that is, the ratio of the resistivity of a rock

saturated with electrolyte solution to the resistivity of the

electrolyte solution.

In the case of a porous plug it is perhaps more useful to consider the equa-tion in terms of permeability reduction rather than of viscosity increase, if we put

k=77LV/P2A and ka =7?LV/PA

where k and kQ

are true and apparent permeabilities, respectively. Then k/ka =

P/P2 and

k/k = 1 + D 2£2/l6 1T

2 k77K x 1/F (14)


or the fractional decrease in permeability is given by

k - ka _ D2£2k 16Tr2k77KF + D 2

£2

Table 1 . - Apparent Permeabilities

(15)

k-k.

£ D K k Fk »'

100 80 10-6 0.53 5.9 1.5

5 2 10-12 0.53 5.9 2.4

100 80 10-6 0.13 11.4 3.1

5 2 10-12 0.13 11.4 4.9

100 80 10-6

0.01 10.0 34.0

5 2 10-12 0.01 10.0 35.3

Table 1 gives percentage reduction in permeability calculated by equation

(15) for selected core permeabilities; the k values are in darcies, F is the forma-

tion factor, and each core is assumed to give a zeta potential of 100 mv in water

and 5 mv in an oil of dielectric constant 2.

EFFECT OF CONDUCTING FILM

Mineral surfaces are more frequently water wet than oil wet, and so it is

interesting to investigate the effect of a thin wetting film on electroviscosity

.

Reconsider the derivation of the electroviscosity equation; it is apparent that the

magnitude of the streaming potential is dependent on the conductivity of the flow-

ing liquid so that for the same zetas the streaming potential is low if conductanceis high, and the streaming potential is high if the conductivity is low.

The streaming potential of a low-conductance liquid (for example, benzene)through a water-wet capillary is less than that expected in the absence of the

water film because the streaming potential is determined by the apparent conduc-tivity of the tube, that is, the conductivity due to the contribution of both liquids.

Consider such a tube of radius R with a water film of thickness 8 = R-r.

Let

Ki = conductivity of water

K2 = conductivity of oil

K3 = conductivity (apparent) of liquids in capillary

For a section of the tube of length L,

Cell constant of water-filled part = L/tt(r2 - r2)

Cell constant of oil-filled part = L/nr 2

Cell constant of whole tube = L/ttR2

Then, if

Measured resistance of water section = P \

Measured resistance of oil section = P2Measured resistance of whole tube = ^3


10 " 10

Water Saturation

Fig. 6. - Effect of water saturation on apparent

viscosity of oil flowing through a 1 -micron

capillary.

l//>3 = !/>!+ I//5

2

and because

1//0J= tt(R2 - r 2)K

1/L; l/p 2

= irr2K2/L; I//O3 = ttR

2K3/L

therefore

K3= r 2/R2

• K2 + (R2 - r 2 )/R2 • Kj

Now water saturation, Sw = tt(R 2 - r 2 )L/VR2 L =

and so K 3= (1 - SW)K2 + SwKj

In terms of thickness of the water layer

K3- K

2- 28/R • (K

2- K

x )

(R< :2 )/R2

and if K„<< K,

or

K3- 28KJ/R

K, SwK l

Figure 6 is a plot of 77 a/77 against Sw showing the rapid decrease of7^ a as

the wetting phase saturation increases. The tube considered has a radius of

0.5 x 10"" 4 cm and exhibits a zeta potential of 30 mv when filled with a hydro-

carbon of D = 2, K2 = 10-12 , Kx- 10 -6 .

TWO- PHASE FLOW

In considering flow of two phases through a capillary tube we will take the

model used by Yuster (1951) of "a single capillary with the non-wetting phaseflowing in a cylindrical portion of the capillary and concentric with it. The wet-

ting phase will flow in the annulus between the capillary wall and the non-wetting

phase"

.


Assuming that both the capillary surface and the oil surface are negatively

charged, the charges at both surfaces are balanced by double layer charges car-

ried in the solution in a Helnholtz double layer at a distance X from the surfaces.

As the ionic concentration in the aqueous phase in contact with both solid and oil

is identical, X is the same for both surfaces.

By first deriving an equation for streaming potential, then for electro-osmosis,

and combining them as was done above, (Street, to be published) it is possible to

obtain expressions for the apparent viscosities in both the oil and the water phases.These expressions are

R^v

'aw1 f

/wi \

D 2(£i

+ S 2£2 ) (£j - S [1 + 4X77 S 2]£ 2 )= 1 + a - y— ^^^_

(16)

'W1 Sw

Z = (1 + S 2) (1 + s - -3)in S

"2

-^ = 1- - ^5o_8ir

2RXT7wK Sw

where £ , and £~ are tne zeta potentials at the mineral and oil surfaces, respec-tively; K is the conductivity of the aqueous phase; rj is the viscosity of the aqueousphase; and 77 that of the oil phase.

There are several points to be considered in the use of these expressions.

First it is assumed that X is small in relation to R-r, that is, the two double layers

in the water must not overlap. In fact, the double layers will affect each other

even at quite large distances of separation. In the simple case of two equal po-tentials, ty , separated by a distance 2h, Elton and Hirschler (1949) show that

m cosh h/X

where \j/m is the potential at a distance h from either surface.

The oil phase will have a dielectric constant much less than the aqueousphase, most likely of the order of 2 rather than 80, and this affects the developmentof potential, the thickness of X , and the concentration of dissolved substance in

the oil phase (Verwey and Overbeek, 1948).

It is perhaps easier to see the effect on X using the approach of Klinkenbergand van der Minne (1958) who show that

K

where EQ = absolute dielectric constant of vacuumAm = coefficient of molecular diffusion

Since A is approximately the same value for both water and a hydrocarbon, then

V^w =V8^ •

80 Ko


Thus the double layer thickness developed in the hydrocarbon phase is muchgreater than that developed in the aqueous phase. The electroviscous effect is

apparent only when the capillary radius is small (1 x 10"^ cm or less) so normal-

ly the double layer thickness in the oil phase would be greater than its radius.

Under these conditions we can assume a homogeneous distribution of charge through-

out this phase and use Rutgers, de Smet, and de Moyer's expression (1957) in

calculating the contribution of this phase to the total streaming potential.

These considerations have been borne in mind in the development of equa-tions (16) and (17), and these expressions should not be used when R-r approachesX and certainly not if R-r< X . In terms of water saturation we can say that the

definite lower limit of applicability is

Sw = 2X/R

Because the movement of the two liquids through a capillary moves positive

charges in the water phase and negative charges in the oil phase, the magnitudeand sign of the streaming potential set up depends on the relative sizes of these

charges, their sign, the viscosity ratios of the liquids, and their saturation in the

tube, that is, on £,, £ ?' ^ ' ^ o' anc^ ^w* Since both positive and negative charges

are transported it is possible for the streaming potential to be either positive or

negative downstream, and the effect on liquid viscosity may be either to increase

it or to decrease it.

It is perhaps appropriate to point out that £ 2is the zeta potential measured

in the water phase against the oil-water interface.

DISCUSSION

It has been shown in the foregoing that viscosity will be increased when aninterfacial charge exists, but the increase will be unimportant unless the charge

is high, the liquid conductivity low, and the flow channel narrow.

Although it could be expected that hydrocarbons would show a large effect

because of their low conductivity, this need not necessarily follow because thin

conducting films in narrow pores will increase the apparent conductivity so as to

considerably reduce the back pressure.

When both water and oil flow through a capillary the interaction of the various

factors may cause the apparent viscosity of either phase to be less than the bulk

viscosity instead of greater. Normally the high ionic concentration of an oil-field

brine will reduce X so as to give such low zeta potentials at both interfaces that

the effect will be negligible. It is possible, however, that the effect could be

significant in a fresh-water flood or, more important, that electrolytes could be

added to the flood water to alter £ , and £~ so as to give maximum recovery.

In the laboratory, core experiments are frequently conducted with both fresh

water and brine in order to determine their relative effects; any low permeability to

fresh water is ascribed to the presence of swelling clays. While admitting that

this is a potent factor in permeability reduction, it is also possible, especially

in relative permeability experiments, that the electroviscous effect is also oper-

ating. The presence of clays will in itself tend to increase the zeta at the water-

mineral interface (Street and Buchanan, 1956) and this may increase the viscosity

even though the clays are of the non-swelling variety.

If the predictions can be borne out experimentally, then we would expect to

find a dependence of relative permeability on ionic concentration in the aqueous

phase during the flow of oil and water through low permeability cores.


At constant surface charge density, increase of ionic concentration decreasesthe zeta potential so that very little charge transport occurs in the water phase;

however, the balancing negative charges in the oil phase will still be carried with

the stream, and the tendency at higher concentrations should be for the oil phaseviscosity to increase while that of the water phase stays relatively constant. How-ever, if the ionic concentration or Sw increases beyond a certain point, back flow

of current through the solution annulus will cause the effect on the oil phase to

become negligible also.

It is hoped to initiate laboratory tests in the near future with the object of

correlating measured zeta potentials at oil-water and mineral-water surfaces with

apparent viscosities of the flowing oil and water phases.

APPENDIX

The identity of \/k with the double layer thickness is by no means obviousand although an understanding of it is not essential to the solution of the problemsinvolved here, nevertheless it would seem more complete to include it. The fol-

lowing is modeled closely on the treatment given by Abramson, Moyer, and Gorin

(1942).

Let equation (5) be written as

y2^ =< 2^ (5a )

Because for a flat surface, or one of large radius of curvature, the potential ^ de-pends only on the distance x from the surface, equation (5a) becomes

and a general solution of equation (5b) is

^r= Ae-*x + Be+ *x (5c)

and because y= when x - coand ty = £ when x = 0, B = and A = £ , so

<j/ = £e -/cx (5d)

Because at any point the potential is the algebraic sum of that due to the

charges on the particle surface and those in the double layer, and if ^^ is the po-tential due to the double layer charges, hence

</>= £ +*/' 1(5e)

Then

Dr

£4 _Q_ d±_W " ' Dr2+

drx=r


and because anywhere inside the sphere and at its surfaced' is constant, thus

dii/—, = and sodr

and because Q = 4ttt ,

$L\dx/x=r " Dr 2

t^which for a particle of any shape has the general form

- Ancf/D (5f)dx/x=o

Differentiation of equation (5d) gives

dx ^

and substitution into equation (5f) at x=o gives

dx

therefore

£ = 4tto/Dk (5g)

Comparison of equations (2) and (5g) shows that X = \/k. It should also be noted

that at any distance x from the surface, the potential \ff is given by


REFERENCES

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Butler, J. A. V., 1940, Electrocapillarity: Methuen, London, p. 93.

Elton, G. A. H., 1948a, Electroviscosity I: The flow of fluids between surfaces

in close proximity: Royal Soc. Proc, v. A194, p. 259.

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Lorenz, P. B., 195 2, The phenomenology of electro-osmosis and streaming po-

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Macauly, J. M., 1936, Range of action of action of surface forces: Nature, v.

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Mukherjee, J. N., 1920, The origin of the charge of a colloid particle and its

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Illinois State Geological Survey Circular 263

18 p., 6 figs., 1 table, 1959

CIRCULAR 263

ILLINOIS STATE GEOLOGICAL SURVEYURBANA

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